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Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions

Year 2016, Volume: 4 Issue: 2, 257 - 265, 01.03.2016

Abstract

In this paper, a matrix method based on Legendre collocation points on interval [-1, 1] is proposed for the approximate solution of some second order nonlinear ordinary differential equations with the mixed nonlinear conditions in terms of Legendre polynomials. The method, by means of collocation points, transforms the differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Legendre coefficients. The numerical results show the effectiveness of the method for this type of equation. When this method is compared with the other usual techniques, results would be easier and have higher accuracy.


References

  • G.F.Corliss, Guarented error bounds for ordinary differential equations, in theory and numeric of ordinary and partial equations (M.Ainsworth, J.Levesley, W.A Light, M.Marletta, Eds), Oxford Universty press, Oxford, pp. 342 (1995).
  • H. Bulut, D.J. Evens, On the solution of Riccati equation by the Decomposition method, Intern. J. Computer Math., 79(1) (2002) 103-109.
  • G. Adomian, Solving frontier problems of physics the decomposition method, Kluwer Academic Publisher Boston, (1994).
  • L.M. Kells, Elemetary Differential Equations, McGraw-Hill Book Company, Newyork, (1965).
  • S.L. Ross, Differential Equations, John Wiley and Sons, Inc. Newyork, (1974).
  • A. Gurler,S. Yalcinbas, Legendre collocation method for solving nonlinear differential equations, Mathematical & Computational Applications, 18 (3) (2013) 521-530.
  • S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation 127, (2002) 195-206.
  • S. Yalcinbas, K. Erdem, Approximate solutions of nonlinear Volterra integral equation systems, International Journal of Modern Physics B 24(32), (2010) 6235-6258.
  • M. Sezer, A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, International Journal of Mathematical Education in Science and Technology 27(6), (1996) 821-834.
  • S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation, 127, (2002) 195-206.
  • D. Tas¸tekin,S. Yalcinbas, M. Sezer, Taylor collocation method for solving a class of the first order nonlinear differential equations, Mathematical & Computational Applications, 18 (3) (2013) 383-391.
  • S. Yalcinbas, D. Tas¸tekin, Fermat collocation method for solving a class of the second order nonlinear differential equations, Applied Mathematics and Physics, 2 (2) (2014) 33-39.
  • D. Tas¸tekin,S. Yalcinbas, Fermat collocation method for solving a class of the first order nonlinear differential equations, Journal of Mathematical Sciences and Applications, 2 (1) (2014) 4-9.
  • H. Gurler, S. Yalcinbas, A new algorithm for the numerical solution of the first order nonlinear differential equations with the mixed non-linear conditions by using Bernstein polynomials, New Trends in Mathematical Sciences, 3 (4) (2015) 114-124.
  • S. Yalcinbas, H. Gurler, Bernstein collocation method for solving the first order nonlinear differential equations with the mixed non-linear conditions, Mathematical & Computational Applications, 20 (3) (2015) 160-173.
  • S. Yalcinbas, K. Erdem Bicer, D. Tas¸tekin, Fermat collocation method for the solutions of nonlinear system of second order boundary value problems, New Trends in Mathematical Sciences, 4 (1) (2016) 87-96.
Year 2016, Volume: 4 Issue: 2, 257 - 265, 01.03.2016

Abstract

References

  • G.F.Corliss, Guarented error bounds for ordinary differential equations, in theory and numeric of ordinary and partial equations (M.Ainsworth, J.Levesley, W.A Light, M.Marletta, Eds), Oxford Universty press, Oxford, pp. 342 (1995).
  • H. Bulut, D.J. Evens, On the solution of Riccati equation by the Decomposition method, Intern. J. Computer Math., 79(1) (2002) 103-109.
  • G. Adomian, Solving frontier problems of physics the decomposition method, Kluwer Academic Publisher Boston, (1994).
  • L.M. Kells, Elemetary Differential Equations, McGraw-Hill Book Company, Newyork, (1965).
  • S.L. Ross, Differential Equations, John Wiley and Sons, Inc. Newyork, (1974).
  • A. Gurler,S. Yalcinbas, Legendre collocation method for solving nonlinear differential equations, Mathematical & Computational Applications, 18 (3) (2013) 521-530.
  • S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation 127, (2002) 195-206.
  • S. Yalcinbas, K. Erdem, Approximate solutions of nonlinear Volterra integral equation systems, International Journal of Modern Physics B 24(32), (2010) 6235-6258.
  • M. Sezer, A method for the approximate solution of the second order linear differential equations in terms of Taylor polynomials, International Journal of Mathematical Education in Science and Technology 27(6), (1996) 821-834.
  • S. Yalcinbas, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation, 127, (2002) 195-206.
  • D. Tas¸tekin,S. Yalcinbas, M. Sezer, Taylor collocation method for solving a class of the first order nonlinear differential equations, Mathematical & Computational Applications, 18 (3) (2013) 383-391.
  • S. Yalcinbas, D. Tas¸tekin, Fermat collocation method for solving a class of the second order nonlinear differential equations, Applied Mathematics and Physics, 2 (2) (2014) 33-39.
  • D. Tas¸tekin,S. Yalcinbas, Fermat collocation method for solving a class of the first order nonlinear differential equations, Journal of Mathematical Sciences and Applications, 2 (1) (2014) 4-9.
  • H. Gurler, S. Yalcinbas, A new algorithm for the numerical solution of the first order nonlinear differential equations with the mixed non-linear conditions by using Bernstein polynomials, New Trends in Mathematical Sciences, 3 (4) (2015) 114-124.
  • S. Yalcinbas, H. Gurler, Bernstein collocation method for solving the first order nonlinear differential equations with the mixed non-linear conditions, Mathematical & Computational Applications, 20 (3) (2015) 160-173.
  • S. Yalcinbas, K. Erdem Bicer, D. Tas¸tekin, Fermat collocation method for the solutions of nonlinear system of second order boundary value problems, New Trends in Mathematical Sciences, 4 (1) (2016) 87-96.
There are 16 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Salih Yalcinbas

Tugce Ulu This is me

Publication Date March 1, 2016
Published in Issue Year 2016 Volume: 4 Issue: 2

Cite

APA Yalcinbas, S., & Ulu, T. (2016). Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions. New Trends in Mathematical Sciences, 4(2), 257-265.
AMA Yalcinbas S, Ulu T. Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions. New Trends in Mathematical Sciences. March 2016;4(2):257-265.
Chicago Yalcinbas, Salih, and Tugce Ulu. “Legendre Collocation Method for Solving a Class of the Second Order Nonlinear Differential Equations With the Mixed Non-Linear Conditions”. New Trends in Mathematical Sciences 4, no. 2 (March 2016): 257-65.
EndNote Yalcinbas S, Ulu T (March 1, 2016) Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions. New Trends in Mathematical Sciences 4 2 257–265.
IEEE S. Yalcinbas and T. Ulu, “Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions”, New Trends in Mathematical Sciences, vol. 4, no. 2, pp. 257–265, 2016.
ISNAD Yalcinbas, Salih - Ulu, Tugce. “Legendre Collocation Method for Solving a Class of the Second Order Nonlinear Differential Equations With the Mixed Non-Linear Conditions”. New Trends in Mathematical Sciences 4/2 (March 2016), 257-265.
JAMA Yalcinbas S, Ulu T. Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions. New Trends in Mathematical Sciences. 2016;4:257–265.
MLA Yalcinbas, Salih and Tugce Ulu. “Legendre Collocation Method for Solving a Class of the Second Order Nonlinear Differential Equations With the Mixed Non-Linear Conditions”. New Trends in Mathematical Sciences, vol. 4, no. 2, 2016, pp. 257-65.
Vancouver Yalcinbas S, Ulu T. Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions. New Trends in Mathematical Sciences. 2016;4(2):257-65.